Theorem 4 (Corollary 4.4 of [1]}) 
Let \((\mathbf {\Phi }_x, \mathbf {\Phi }_u, \mathbf {\Delta })\)  be a solution to
\(\begin{bmatrix} zI-A & -B \end{bmatrix} \begin{bmatrix} \mathbf {\Phi }_x \\ \mathbf {\Phi }_u \end{bmatrix} = I + \mathbf {\Delta }, \quad \mathbf {\Phi }_x, \mathbf {\Phi }_u \in \frac{1}{z}\mathcal {R}\mathcal {H}_\infty .\) 
